Metamaterials: myth or reality? "Reverse" refractive index. Part 1
1.1. History of metamaterials concept development
Theoretically the opportunity of negative refractive index mediums existence was offered by the Soviet physicist Victor Veselago as far back as in 1967 .
Veselago is one of the first authors who began to develop the idea concerning materials with anomalous properties which were later named as "metamaterials".
It is necessary to mark that Veselago only made a hypothesis that materials with anomalous properties, such as the negative refractive index (Negative Index Materials, NIMs) and others can exist. There were no facts confirming this hypothesis.
In view of materials absence with the anomalous properties the hypothesis was forgotten up to the 21st century.
Surge of interest in the negative refraction of electromagnetic waves including light, when the refracted beam deviates on other side of a normal to media border, arose at the beginning of the 21st century after publications of San Diego University (USA) scientists group which reported about creation of the composite materials having the negative index of refraction [2–4].
Also it is necessary to note the article of English physicist J. Pendry  who made the following prediction: a diffraction limit at the size of a focal spot peculiar to conventional lenses in parallel-sided NIM plate won’t appear. The similar focusing device was called a perfect lens by Pendry. The peculiarity of such perfect lens is that the distance between a NIM lens and research subject shall be less than light wavelength in the visible range, i. e. ~50–200 nm. According to Pendry, it means that it is possible to create optical microscopes with a resolution never before available (it is impossible to distinguish two subjects by means of an optical system if they are at distance less than observation wavelengths).
A number of the researches and publications devoted to NIMs grow relating to supposed possibilities of their application.
1.2. Present situation
However publications on NIMs still cause disputes at seminars and on pages of academic periodicals where both history of origin of this trend, and the principal ideas serving as its basis are discussed.
Disputes arise because nobody managed to create NIM working in visible spectrum. Also nobody could explain physical mechanisms of supposed abnormal phenomena.
A paradox has arisen: there are theoretical predictions, at that publications and researches exist on a large scale, but there are no explanations of the physical mechanism of supposed surprising properties, as well as NIMs working in the visible range.
Let us see in more detail, what characteristics hypothetical NIMs should have according to Veselago. And NIMs truly have the following surprising properties (effects):
• reverse (negative) refractive index;
• Doppler’s reverse effect;
• Vavilov-Cherenkov reverse effect;
• effect of the "cap of darkness’ [6, 7];
• effect of "superlens’ (perfect lens) ;
• the negative magnetic and dielectric inductive capacity (chapter 2.5);
• phase and group rates of electromagnetic waves have opposite directions in NIMs (chapter 2.9);
• Goos-Hanchen effect (chapter 2.12).
Let us give the general definition of metamaterial (NIM).
Metamaterial is such a composite material the features of which are less likely caused by individual physical properties of its components, than by its microstructure.
Main methods of metamaterial preparation are the following:
• embedding of nano-, submicron or micron objects in the transparent polymeric medium.
• creation of the array consisting of plates. The ring-shaped elements (closed and nonclosed) executed from metal are applied on the plates. Other components can be applied above ring-shaped elements.
• metamaterial creation by the method of electron beam lithography (EBL).
• photonic crystal creation (chapter 2.10).
Such composite material shows features that are not typical for objects occurring in nature. Let us try to find out how it can be possible.
1.3. Negative refractive index
One of most discussed effects in recent times is the negative refractive index.
Let us dwell upon this effect as other characteristics of NIMs are also caused by negative refractive index.
Let us consider more attentively what Victor Veselago has assumed in 1967 . Veselago’s article has special significance as all subsequent authors’ works are based on his work and use his drawings and conclusions.
The analysis of drawings offered by Veselago (fig. 1a, b) to which all further authors refer, raises some doubts.
• Why did the light beam deviate only on the phase border, and does not deviate when passing through the composite saturated with any objects, elements (fig. 1a)?
• Why is there no refraction because of changing medium density in fig.1a, and why inlet and outlet angle are different (ϕ и ψ) in fig.1b?
• Why did the beam just follow the trajectory 3 in "the left medium" (fig. 1b)?
• Why is the beam reflected from the left substance absent? There is neither explanation, nor proofs for this fact.
Light refraction is shown in fig. 2  at passing from air to denser medium according to a Snell’s law (fig. 2a) and the refraction at passing through NIM is shown in fig. 2b. Apparently, no updates had yet been introduced over a period of 40 years, except for a reflected beam and wave vector direction of k = 2 π/λ occurrence.
1.4. Discussion of supposed effect of reverse refraction
The structure of NIMs and experiments described in articles [2–4, 7, 8] demonstrate that the electromagnetic radiation can follow nontypical trajectory, after double reflection (fig. 2b), for example. Apart from reflection it is necessary to consider diffraction, diffusion, refraction through an object and the object size (Rayleigh’s theory, Mie’s theory, and the theory of Fraunhofer), refractive index, geometry, roughness, object’s dielectric capacity, plasmon enhancement, creation of plasmon polaritons, creation of evanescent field, effect of Bragg diffraction, dispersion and light absorption (see chapter 2 in more detail). At this stage we will consider an interaction of electromagnetic radiation with an object in simplified form, being guided only by reflection, refraction and interference.
Instead of introduction of nano-, submicron or micron objects in the transparent medium NIM can be made, for example, of metal (Me) objects of 5–15 mm, counting on microwave radiation (super-high-frequency radiation) . The microwave radiation will be reflected from metal objects. The common principle of NIM structure creation remains changeless, i. e. reflection of electromagnetic radiation from an obstacle with the subsequent interference takes place. More detailed analysis of R. Shelby’s article with co-authors  is given below in the text.
It is also possible to bisect electromagnetic radiation beam, using the effect of double refraction, for example, affecting material with electric field (the Kerr effect, see chapter 2.11 for more detail). But Kerr effect does not influence electromagnetic radiation, but changes the structure of material, orienting molecules under the influence of electric field.
Thus, the mechanism of electromagnetic radiation interaction with an object (objects) followed by the effect of the reverse refraction can be realized in several ways:
• reflection (with the subsequent interference) of visible light from objects nano-, submicron and micron size;
• reflection of the microwave radiation from metal objects the sizes of which are 5–15 mm;
• double refraction under the influence of an electric field (electrooptical Kerr effect).
Let us try to find out what is meant here and whether it is practical to obtain and use NIM. Or is it only the next "soap bubble", like prospects of hydrogen power engineering or danger of ozone layer exhaustion?
The model showing how electromagnetic radiation is capable to be reflected in NIM at the reverse angle according to fig. 1b and fig. 2b (trajectory 3) offered by Veselago is presented in fig. 3.
The reverse refraction in NIM represents not the abnormal refraction, but the effect of electromagnetic radiation reflection of the objects which are in the transparent medium. For example, it can be the superfine aluminum reflecting film being in composition on the basis of polyvinyl alcohol (the basis for manufacture of film polarizer).
Electromagnetic radiation gets inside the transparent medium, undergoing a classical "positive" refraction on media border, according to a Snell›s law (fig. 2a), then it is reflected from a film surface, and easily passes through the polymer mass (fig. 3a), and is reflected again at the output (according to fig. 1a and 2b) and gets to the observer.
To create a lens from such object, the following aspects are necessary:
• the particular size and geometry of film arrangement in a composite;
• optimum distance between a NIM film and an object of a research (fixed angles of light exit);
• monochromatic light;
• interference signal amplification;
• manufacture accuracy.
The geometry of reflecting objects incorporated in surface layers of transparent polymer can be different (not only a tape). For example, as the reflecting element nanocellulose crystals, having the shape of the extended parallelepiped, either spherical metal particle, or crystals in the form of pentagon dodecahedrons (dodecahedrons) can be used.
What should be the size of reflecting objects containing in a composite? It is possible to consider 3 types of objects: 1 are objects the size of which exceeds the incident light wavelength λ by over ten times, 2 are objects comparable with λ sizes, 3 are objects which is much less than λ. We will carry out the assessment proceeding from impinging light wavelength of 450 nm (visible range).
1. The size of objects, entered into the transparent medium, should be more than 4.5 μm (d >> λ) to make geometrical optics laws described by Fresnel formulas work. These objects should have a good reflectivity. Since reflection laws shall prevail over refraction laws, it is expedient to make instead of micron objects introduction the NIM in the form of a reflector with a set of concentric circles of the reflecting superfine nanoribbon. The photo of similar ring structure is provided on fig. 4 .
2. In case of object sizes closeness to λ of impinging light (d ≈ λ) the scattering direction diagram becomes the composite (see chapters 2.1 Mie’s theory of scattering and 2.2 Fraunhofer’s theory of diffraction for more detail). The wave interference reflected from various sites of object surface is shown. The intensity of light scattering at particular angle depends on that how many times does the wave settle on object diameter and therefore scattering intensity depends heavily on the object sizes. The supposed size of objects which are placed in a composite should be of 100–1 000 nm.
The number of reflecting submicron objects on composite surfaces should be such that the light beam will be reflected no more than two times (at input and at output). Otherwise instead of NIM there should be an ordinary semitransparent dissipate light ribbon (there will be a multiple scaterring and absorption).
3. In case if λ is much more than the size of objects contained in material (d << λ), a Rayleigh scattering (as well as molecular) take place. In that case the question concerns already nanoobjects with sizes less than 45 nm. So small objects are not of interest from the point of view of NIMs creation working in a visible range (the light beam does not change a trajectory, and partially scatter, loses energy and respectively increases a wavelength).
Thus, our proposed scheme of light passing through NIM (fig. 3a) can be a prototype of a plain superlens in a visible range.
• The concept of the negative refraction and a number of formulas with negative value only confuses and diverts from comprehension of process content . It is more logical to use the term "reverse refraction".
• Creation of the microscope several times exceeding the optical microscope diffraction limit (~300 nm) on the basis of NIM is irrelevant as the following devices exist and are regularly used:
• near-field microscope (resolution is up to ~25 nm),
• atomic-force microscope (resolution is up to ~2 nm),
• electronic microscope (resolution is up to ~1 nm).
• The fundamental restriction, certainly particular to NIM (dispersing composite) is the losses caused by electromagnetic radiation scattering on the incorporated objects.
Typical scattering is followed by additional effects, such as resonance fluorescence, Raman (combinational) scattering and parametrical light scattering .
Qualitative advance will be reached when it is succeeded in creating NIM with small losses in a visible range. There is a dilemma which means that on the one hand it is necessary for the light to be reflected from objects, and on the other hand it is necessary to provide small losses (and distortions) when NIM passing.
• In his work M. Stockmann claims that superresolution obtaining in optical range is impossible . "If resonance inclusions are entered into the passive dielectric medium, the analysis of the ratios based on causality principle (Kramers-Kronig relation) shows that the negative refraction is always followed by essential losses".
The causality principle (in physics) sets tolerance limits of physical events influence against each other.
Kramers-Kronig relation is the expression of dispersion ratios in a classical electrodynamics. In this case
ε (ω) = ε′ (ω) + iε″ (ω),
where ε (ω) is a permittivity, ω is an electromagnetic frequency.
The real and imaginary parts of permittivity (respectively they are the 1st and the 2nd members of equation) define refractive index and absorption index (optical constants) of this medium. Thus, these indexes are not independent of each other, and, therefore, the principal possibility occurs to calculate on a range of one of optical constants occurs to calculate a range of another one, without resorting to direct measurements of the last one . In some cases it allows it allows reducing the volume of experimentally obtained information in some cases which is necessary for defining the optical constants, for example, in the region of intensive absorption bands of the condensed media. Feasibility of Kramers-Kronig relations was repeatedly checked experimentally for various media in various aggregative states and at different temperatures (crystals, liquids, solutions) .
It is also claimed in works [15, 16] that the negative refraction contradicts the causality principle and therefore it does not exist.
• The authors of articles [17, 18] make critical observations with respect to Pendry’s prediction for a superlens. It should be noted that this criticism remained almost unnoticed because of a large number of theoretical works in favor of Pendry.
• Let us consider the question concerning refractive index n ≤ 1. What will be the electromagnetic radiation speed in the medium with such refractive index?
It is known that refractive index is directly linked with electromagnetic radiation velocity in the medium. So, for vacuum n = 1.0, for air n = 1.000292, for water n = 1.334, for diamond n = 2.42. For example, the velocity of electromagnetic radiation in water will be 0.74 . c, i. e. 2.24 . 108 m/c.
In the work of V. Shalaev and his co-authors  the negative refractive index of n = – 0.3 is achieved. A similar situation is considered in Veselago’s work , n = – 1.
How to interpret n = – 0.3? It turns out that electromagnetic radiation velocity in NIM will exceed the light velocity more than three times.
It appears that researchers have passed from practical physics into the field of the abstract mathematics without being noticed by most readers. The refractive index n ≤ 1 describes the phase velocity of electromagnetic radiation (abstract velocity connection between oscillation phases in various space points), but not a group velocity (actual), see chapter 2.9. And a phase velocity can be of any value, including the value that is more than light velocity.
• The work of Shelby and his co-authors "The experimental verification of negative refractive index", 2001 is regarded as one of the main proofs of NIM existence  and accuracy of Veselago’s hypothesis.
Let us consider this work in more detail.
Researchers succeed in obtaining NIM at sample treatment with microwave radiation with the wavelength of 3 cm (fig. 5) (a microwave spectral range is 0.3–30 cm). At the that ring-shaped resonators were used for obtaining the negative magnetic conductivity and wire elements were used for receiving the negative permittivity.
Actually it means that experimenters received the result by introduction of a large number of the multiplex objects into material and creation of electromagnetic fields in the sample (fig. 5a). As a result they managed to direct artificially (to reflect) a beam in nontypical direction. It is only the attempt to adjust an experiment to a hypothesis.
The analogy with double refraction effect in anizotropic crystal is considered in work . At double refraction (see chapter 2.11) an unusual beam follows the abnormal trajectory. The beam also follows an abnormal trajectory owing to a multiple reflection of the microwave radiation from metal elements of NIM structure in Shelby’s work. As a result one of the prevailing directions of scattered and reflected radiation coincides with the supposed direction of the reverse refraction.
It should be noted that the experiment made by Shelby and his co-authors corresponds to static scattering method (scattering of Fraunhofer, chapter 2.2) except for the fact that authors do not adduce the intensity of angular radiation distribution at output from NIM.
Authors of the article use such concepts as the negative permittivity and the negative magnetic permeability (theoretical assumption by Veselago). However, there is no physical explanation for these negative values. Authors do not say anything how they have measured them.
The fact that researchers guide microwave radiation with the wavelength of 3 cm to a gap between Al disks of 1.2 cm wide that leads to deformation of the entering beam before an entrance to a cellular structure of NIM can be represented as technical shortcomings of experiment.
It should be noted the fact that material with effect of the reverse refraction will work only at a certain at the particular wavelength depending on geometry of a cellular structure of a sample that is confirmed by the data presented in article .
Microwave radiation instead of visible light is used for the experiment. It can be explained with the fact that strong microwave radiation, unlike visible light, easily passes through nontransparent dielectric materials. Besides, the use of microwaves reduces the requirements for miniaturization of a structure, enabling to manufacture plates of corresponding size (radiation wavelength and anisotropic objects of NIM should have the comparable sizes).
As a result of microwave radiation influence in the studied NIM electromagnetic fields are induced by means of copper structural elements (similar to energized metal objects subjects’ placement into microwave oven, current flows through the object inducing electromagnetic field and causing sparking). The high intensity electromagnetic field in ladder-type of NIM (fig. 5b) reflects a part of microwave radiation in the direction corresponding to a hypothetical reverse refraction (according to a trajectory of a beam 3 in fig.1b).
NIM geometry in the form of a prism (fig. 5b) is not casual. A part of beams going through cellular structure of NIM diagonally from left to right (because of reflection and scattering on shielding disks), will be reflected in "ladder" and at output from NIM should move from right to left.
Shelby’s experiment was improved in work . A multilayer cellular dielectric-metal structure, subjected to infrared radiation in the range of 1200–1700 nm was used as NIM (the alternating layers of Ag 30 nm thick and layers of MgF2 50 nm thick). NIM is also executed in the form of a prism, the path of beams corresponds to fig. 5b. Actually, there is a reflection of infrared radiation at output from NIM prism because of a cellular structure.
Summarizing information explained in work , it is possible to state the following.
• Authors of the article have honestly investigated microwave radiation passing through cellular multiplex material.
• A part of the microwave radiation really deviated in the direction of reverse refraction.
• A deviation of the microwave radiation occurred due to reflection from metal elements introduced in material. This fact does not demonstrate refraction.
• The microwave radiation differs significantly from visible light according to penetrating properties that allows claiming that this experiment cannot be miniaturized and repeated for visible light. Besides the authors of the article honestly notice: "It is unlikely that the inherent material properties of conductors will scale much past the infrared, rendering left-handed materials, such as those used here, ineffective".
• Scattering and absorption on the objects introduced in NIM should neutralize all efforts on creation of a lens from NIM in the visible range area .
• Pendry’s prediction for a superlens remains only a prediction. It is premature to make assumptions that the effect of a superlens should work in a near field (at distances less thаn λ of visible light, i. e. ~50–200 nm).
The question arises concerning whether it is possible to make metamaterial or not. To answer this question, it is necessary to deal with a number of the complicated optical effects, to separate courageous forecasts of a number of scientists from really observed optical effects. First of all it is necessary to deal with a physical essence of observed effects, but not with their mathematical description or mathematical modelling. We will try to answer the above question and we will consider optical effects in the following issue.
"Metamaterial" can be applied to many modern composite materials, therefore NIM which means material with the negative refractive index will be used further. The terms "left-handed material" (LHM) or "the left media" are synonymous to NIM. "Right media" means material working under classic laws of optics.
 "Metamaterial" can be applied to many modern composite materials, therefore NIM which means material with the negative refractive index will be used further. The terms "left-handed material" (LHM) or "the left media" are synonymous to NIM. "Right media" means material working under classic laws of optics.
 It should be noted that near-field microscopes can overcome the diffraction limit. For this purpose a research object and the scanning probe should be separated by a distance less than λ of radiating laser. The near-field microscope operating principle is not bound to the concept of metamaterials and the negative refraction. See chapter 2.5 and 2.6 for more detail.
 Photonic crystal is a material the arrangement of which is characterised by periodical change of refraction index. For details, see chapter 2.10.
 Dispersion relationships are integral representations of response functions describing a response of balanced stationary physical system to external effects. In the narrow sense dispersion relationships connect the refraction of waves extended in the system with their absorption.